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Theorem alrimii 32423
Description: A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
alrimii.1  |-  F/ y
ph
alrimii.2  |-  ( ph  ->  ps )
alrimii.3  |-  ( [. y  /  x ]. ch  <->  ps )
alrimii.4  |-  F/ y ch
Assertion
Ref Expression
alrimii  |-  ( ph  ->  A. x ch )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)

Proof of Theorem alrimii
StepHypRef Expression
1 alrimii.1 . . 3  |-  F/ y
ph
2 alrimii.2 . . . 4  |-  ( ph  ->  ps )
3 alrimii.3 . . . 4  |-  ( [. y  /  x ]. ch  <->  ps )
42, 3sylibr 217 . . 3  |-  ( ph  ->  [. y  /  x ]. ch )
51, 4alrimi 1975 . 2  |-  ( ph  ->  A. y [. y  /  x ]. ch )
6 nfsbc1v 3275 . . 3  |-  F/ x [. y  /  x ]. ch
7 alrimii.4 . . 3  |-  F/ y ch
8 sbceq2a 3267 . . 3  |-  ( y  =  x  ->  ( [. y  /  x ]. ch  <->  ch ) )
96, 7, 8cbval 2127 . 2  |-  ( A. y [. y  /  x ]. ch  <->  A. x ch )
105, 9sylib 201 1  |-  ( ph  ->  A. x ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450   F/wnf 1675   [.wsbc 3255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-sbc 3256
This theorem is referenced by: (None)
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